\section{Note to Rolf and Simon}
All references to Bj\"ork are to the third edition of `Arbitrage Theory in Continuous Time'.

\section{Q1: Quanto put}
\subsection{Q.1a}
Let $\tilde{r}_J = r_J - \sigma_X^*\sigma_J$. Then we have
\begin{align}
F^{QP}(t,s) & = E^{\mQ}_{t,s}\brackk{e^{-r_{US}\paren{T-t}}Y_0\paren{K-S_J(T)}^+} \nonumber\\
& = e^{-r_{US}\paren{T-t}}e^{\tilde{r}_J\paren{T-t}}Y_0E^{\mQ}_{t,s}\brackk{e^{-\tilde{r}_J\paren{T-t}}\paren{K-S_J(T)}^+}.\label{eq:bs:jpy}
\end{align}
Now, using the pattern-recognition hint, we see that the term 
\begin{align}
E^{\mQ}_{t,s}\brackk{e^{-\tilde{r}_J\paren{T-t}}\paren{K-S_J(T)}^+}
\end{align}
in~\eqref{eq:bs:jpy} is simply the price of a put on $S_J$ in a model in which the risk-free rate is $\tilde{r}_J$ and the volatility is $\sigma_J$. Then, applying Put-call parity, see Bj\"ork proposition 9.2, say, we have
\begin{align}
F^{QP}(t,s) &= e^{-r_{US}\paren{T-t}}e^{\tilde{r}_J\paren{T-t}}Y_0\paren{e^{-\tilde{r}_J\paren{T-t}}K\Phi (-d_2) - s\Phi (-d_1(t,s))}\nonumber\\
& = e^{-r_{US}\paren{T-t}}Y_0\paren{K\Phi (-d_2) - e^{\tilde{r}_J\paren{T-t}}s\Phi (-d_1(t,s))},
\end{align}
where 
\begin{align}
d_{1/2}(t,s) = \frac{\log (s/K) + \paren{\tilde{r}_J \pm \half\lVert \sigma_J\rVert^2}\paren{T-t}}{\sqrt{T-t}\lVert \sigma_J\rVert}
\end{align}
Furthermore, using the pattern-recognition hint, 
\begin{align}
\pdiff{F^{QP}(t,s)}{s} &= e^{-r_{US}\paren{T-t}}e^{\tilde{r}_J\paren{T-t}}Y_0\pdiff{}{s}\paren{e^{-\tilde{r}_J\paren{T-t}}K\Phi (-d_2(t,s) - s\Phi (-d_1(t,s))}\nonumber\\
& = e^{\paren{\tilde{r}_J-r_{US}}\paren{T-t}}Y_0\paren{\Phi \paren{d_1(t,s)}-1}
\end{align}
where we without further proof used the fact delta of a put is $\Phi (d_1(t,s))-1$ (which again follows from Put-call parity).
\subsection{Q.1b}
We consider the portfolio with the dynamics
\begin{align}
V^h(t) = h_1(t)S_J(t)X(t) + h_3(t)B_{US}
\end{align}
where 
\begin{align}
h_1(t) = \frac{g(t, S_J(t)}{X(t)}
\end{align}
and
\begin{align}
h_3(t)B_{US} = V^h(t) - h_1(t)S_J(t)X(t).
\end{align}
See~\figref{fig:hedges} ) (left column) for the performance of several simulations of this portfolio. The experiment was performed with a hedge frequency varying from 200 to 6000 in steps of 200. For each hedge frequency 1000 simulations of the hedging portfolio were performed. The following parameter values were used
\begin{align}
r_{US} &= 0.02\nonumber\\
r_J &= 0.00\nonumber\\
\sigma_X^* &= (0.05, 0.05)\nonumber\\
\sigma_J^* &= (0, 0.2)\nonumber\\
\end{align}
\subsection{Q.1c}
The portfolio og Q.1b is amended with 
\begin{align}
-\Delta^{QP}\paren{t,S_J(t),X(t)}S_J(t)
\end{align}
of currency, but still made self-financing via the domestic bank-account.
The portfolio is thus
\begin{align}
V^h(t) = h_1(t)S_J(t)X(t) + h_2(t)B_J(t)X(t) + h_3(t)B_{US}
\end{align}
where
\begin{align}
h_2(t) = -\Delta^{QP}\paren{t,S_J(t),X(t)}S_J(t)
\end{align}
and
\begin{align}
h_3(t) B_{US}= V^h(t) - \Delta^{QP}S_J(t)X(t) + \Delta^{QP}S_J(t)X(t).
\end{align}
The performance of this portfolio is also given in~\figref{fig:hedges} (right column). As before the experiment was performed with a hedge frequency varying from 200 to 6000 in steps of 200. For each hedge frequency 1000 simulations of the hedging portfolio were performed. The parameter values of Q.1b were used.
\subsection{Comment on the performance of the code}
The implementation of the replication strategy does seem to support the conclusion of the following question (Q.1d), albeit with the following problem: It consistently undershoots compared to the pay-off function. Unfortunately we have not been able to determine the cause of the phenomena, however we suspect it to be related to the initialization of the portfolio.
\subsection{Q.1d}
The intuition to gather from the previous two exercises is the following: The portfolio in question 1b correctly hedges changes in the value of the option due to changes in the stock price, but it does not take into account changes due to fluctuations in the exchange-rate. This is amended by expanding the portfolio by shorting the amount of currency specified in question Q.1c. In general the replicating portfolio is given by
\begin{align}
V^h(t) = h_1(t)S_J(t)X(t) + h_2(t)B_J(t)X(t) + h_3(t)B_{US}(t)
\end{align}
with the restraint that $V^h = F^{QP}$ for all $t$.
This leads us to the following
\begin{align}
\pdiff{V^h}{s} = h_1(t)X(t) = \pdiff{F^{QP}}{s}.
\end{align}
Hence we have 
\begin{align}
h_1(t) = \frac{\pdiff{F^{QP}}{s}}{X(t)} = \Delta^{QP}.
\end{align}
As noted and observed, this does not make the portfolio neutral to changes in the exchange-rate. This is done by considering
\begin{align}
\pdiff{V^h}{x} = h_1(t)S_J(t) + h_2(t)B_J(t) = \pdiff{F^{QP}}{x} = 0,
\end{align}
where the last equality follows since $F^{QP}$ does not depend on $X$. Hence
\begin{align}
h_2(t)B_J(t) = -h_1(t)S_J(t) = -\Delta^{QP}S_J(t).
\end{align}
Finally let
\begin{align}
h_3(t) = \frac{F^{QP} - \Delta^{QP}S_J(t)X(t) + \Delta^{QP}S_J(t)X(t)}{B_{US}(t)}.
\end{align}
Then by theorem 8.5 and theorem 13.7 of Bj\"ork $V^h$ replicates the claim, and it is self-financing.
\def\myTextWdith {0.4}
\begin{figure}
        \centering
        \begin{subfigure}[b]{\myTextWdith\textwidth}
                \includegraphics[width=\textwidth]{problem1_IncorrectHedge}
                \caption{Simulated pay-off of incorrect hedge portfolio}
                \label{fig:gull}
        \end{subfigure}%
        ~ %add desired spacing between images, e. g. ~, \quad, \qquad etc.
          %(or a blank line to force the subfigure onto a new line)
        \begin{subfigure}[b]{\myTextWdith\textwidth}
                \includegraphics[width=\textwidth]{problem1_correctHedge}
                \caption{Simulated pay-off of correct hedge portfolio}
                \label{fig:tiger}
        \end{subfigure}
        
%        ~ %add desired spacing between images, e. g. ~, \quad, \qquad etc.
%          %(or a blank line to force the subfigure onto a new line)
        \begin{subfigure}[b]{\myTextWdith\textwidth}
                \includegraphics[width=\textwidth]{hedge_Error_IncorrectPortfolio}
                \caption{Log of hedge error of incorrect replicating portfolio}
                \label{fig:mouse}
        \end{subfigure}
        ~
        \begin{subfigure}[b]{\myTextWdith\textwidth}
                \includegraphics[width=\textwidth]{hedge_Error_CorrectPortfolio}
                \caption{Log of hedge error of correct replicating portfolio}
                \label{fig:mouse}
        \end{subfigure}
        
        \begin{subfigure}[b]{\myTextWdith\textwidth}
                \includegraphics[width=\textwidth]{std_Incorrect}
                \caption{Standard deviation of hedge error of incorrect replicating portfolio}
                \label{fig:mouse}
        \end{subfigure}
        ~
        \begin{subfigure}[b]{\myTextWdith\textwidth}
                \includegraphics[width=\textwidth]{std_Correct}
                \caption{Standard deviation of hedge error of correct replicating portfolio}
                \label{fig:mouse}
        \end{subfigure}
        
				\begin{subfigure}[b]{\myTextWdith\textwidth}
                \includegraphics[width=\textwidth]{Log_std_Incorrect}
                \caption{Log-Standard deviation of hedge error of incorrect replicating portfolio}
                \label{fig:mouse}
        \end{subfigure}
        ~
        \begin{subfigure}[b]{\myTextWdith\textwidth}
                \includegraphics[width=\textwidth]{Log_std_Correct}
                \caption{Log-Standard deviation of hedge error of correct replicating portfolio}
                \label{fig:mouse}
        \end{subfigure}
        \caption{Quanto-put replication.}\label{fig:hedges}
\end{figure}
\newpage
